Mixed formulation and structure-preserving discretization of Cosserat rod dynamics in a port-Hamiltonian framework

TL;DR

This work develops a mixed, infinite-dimensional port-Hamiltonian framework for the nonlinear dynamics of spatial Cosserat rods with large rotations. By employing a singularity-free director parametrization, it achieves a constant mass matrix and a natural time-differentiated compliance, enabling inextensibility and shear-rigidity constraints to be imposed within the PH setting. The authors then derive a structure-preserving mixed finite-element discretization in space and an energy-momentum consistent implicit midpoint time integrator, guaranteeing exact discrete power balance and angular-momentum conservation. Extensions to visco-elasticity via a generalized-Maxwell model and actuation through pneumatic chambers or tendons are seamlessly integrated into the same PH framework. Numerical examples (including a flying spaghetti, cantilever oscillations, and soft robotic-arm maneuvers) demonstrate energy-momentum consistency, accurate kinematic relations, and the framework’s potential for control-oriented, energy-aware simulations of flexible rods with finite rotations.

Abstract

An energy-based modeling framework for the nonlinear dynamics of spatial Cosserat rods undergoing large displacements and rotations is proposed. The mixed formulation features independent displacement, velocity and stress variables and is further objective and locking-free. Finite rotations are represented using a director formulation that avoids singularities and yields a constant mass matrix. This results in an infinite-dimensional nonlinear port-Hamiltonian (PH) system governed by partial differential-algebraic equations with a quadratic energy functional. Using a time-differentiated compliance form of the stress-strain relations allows for the imposition of kinematic constraints, such as inextensibility or shear-rigidity. A structure-preserving finite element discretization leads to a finite-dimensional system with PH structure, thus facilitating the design of an energy-momentum consistent integration scheme. Dissipative material behavior (via the generalized-Maxwell model) and non-standard actuation approaches (via pneumatic chambers or tendons) integrate naturally into the framework. As illustrated by selected numerical examples, the present framework establishes a new approach to energy-momentum consistent formulations in computational mechanics involving finite rotations.

Figures (16)

• Figure 1: Schematic depiction of a Cosserat rod configuration.
• Figure 2: Schematic rheological model for the generalized-Maxwell approach.
• Figure 3: Schematic depiction for the actuation of a Cosserat rod, where the $k$-th actuation element and related kinematic quantities are displayed.
• Figure 4: Flying spaghetti: Snapshots of the kayak-rowing motion with azimuth and elevation perspective angles $(55,15)$ and the colormap for time $t_{n}$ with $\tikzsetnextfilename{redgrad_bar} \in [0, 15]$.
• Figure 5: Flying spaghetti: Total energy \ref{['legend_total_energy']}$\hat{H}(\bm{x}_{n})$, input work \ref{['legend_ext_work']}$W^{\mathrm{ext}}_{n}$, total energy increments \ref{['legend_energy_inc']}$\hat{H}(\hat{\bm{\bm{x}}}_{n+1})-\hat{H}(\hat{\bm{\bm{x}}}_{n})$, and energy balance violation \ref{['legend_power_balance']}$\Delta E_{n}$.

Theorems & Definitions (10)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 4.1
• Remark 4.2
• Remark 5.1
• Remark 5.2
• Remark 5.3
• Remark D.1